Category:Method of Infinite Descent

This category contains pages concerning Method of Infinite Descent:

There is believed to be a mistake here, possibly a typo. In particular: This technique only works if you can prove that there exists some $n_\gamma$ such that $0 \le n_\gamma < n_\alpha$ for which $\map P {n_\gamma}$ does not work. Otherwise, technically speaking, the proof by Fermat in the historical note is not actually an example of this proof technique. I challenge the student to come up with a proof which genuinely uses the technique as presented here, because I am not sure that it makes logical sense. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by reviewing it, and either correcting it or adding some explanatory material as to why you believe it is actually correct after all. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Mistake}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Let $\map P {n_\alpha}$ be a propositional function depending on $n_\alpha \in \N$.

Let $\map P {n_\alpha} \implies \map P {n_\beta}$ such that $0 < n_\beta < n_\alpha$.

Then we may deduce that $\map P n$ is false for all $n \in \N$.

That is, suppose that by assuming the truth of $\map P {n_\alpha}$ for any natural number $n_\alpha$, we may deduce that there always exists some number $n_\beta$ strictly less than $n_\alpha$ for which $\map P {n_\beta}$ is also true, then $\map P {n_\alpha}$ cannot be true after all.

This technique is known as the method of infinite descent.

The process of deducing the truth of $\map P {n_\beta}$ from $\map P {n_\alpha}$ such that $0 < n_\beta < n_\alpha$ is known as the descent step.